On energy harvesting from ambient vibration

https://doi.org/10.1016/j.jsv.2005.10.003Get rights and content

Abstract

Future MEMS devices will harvest energy from their environment. One can envisage an autonomous condition monitoring vibration sensor being powered by that same vibration, and transmitting data over a wireless link; inaccessible or hostile environments are obvious areas of application. The base excitation of an elastically mounted magnetic seismic mass moving past a coil, considered previously by several authors, is analysed in detail. The amplitude of the seismic mass is limited in any practical device and this, together with the magnitude and frequency of the excitation define the maximum power that can be extracted from the environment. The overall damping coefficient (part of which is mechanical) is associated with the harvesting and dissipation of energy and also the transfer of energy from the vibrating base into the system. It is shown that net energy flow from the base through the damper is positive (negative) for ω>ωn(ω<ωn), but is zero when ω=ωn. The mechanical part of the damper cannot contribute more power than it dissipates and is neutral, at best, when ω/ωn. Maximum power is delivered to an electrical load when its resistance is equal to the sum of the coil internal resistance and the electrical analogue of the mechanical damping coefficient, which differs from what has been claimed. A highly damped system has the advantage of harvesting energy over a wider band of excitation frequencies on either side of the natural frequency, is smaller, but will harvest marginally less power. One possible strategy for variable amplitude excitation is proposed.

Introduction

There has been much recent interest in the concept of low-power microelectromechanical systems (MEMS) that are able to scavenge, or harvest, energy from their operating environment. For example, one can envisage an autonomous condition monitoring sensor, measuring vibration level, being powered by the very same vibration, and transmitting data over a wireless link. Assuming the available power to be small, one envisages that the sensor would need to be intelligent—harvesting and storing energy for a period of time, before taking and transmitting a reading, followed by a further period of harvesting. Such devices could be employed in potentially hostile or inaccessible environments, and would require little or no maintenance.

A mechanical model of such a device, consisting of the base excitation of an elastically mounted seismic mass has received much attention, and was first proposed by Williams and Yates [1]. While the elastic mount is typically modelled as a coil spring, in practice a cantilever beam would provide both the flexibility and necessary constraint of the seismic mass without the need for bearing surfaces which are prone to high friction and hence heat generation at small scales. The means of energy extraction may consist of a magnetic seismic mass moving past a coil, or the beam may be fabricated from piezoelectric material. As far as the mechanical vibration is concerned, the combination of mechanical and electrical damping is treated as equivalent to linear viscous damping (proportional to velocity). This is a fair representation for the electromagnetic conversion considered by Williams and Yates [1], less so for a piezoelectric conversion; nevertheless, many interesting conclusions may be drawn from this simple model. Prototype devices have been described by El-Hami et al. [2], Roundy et al. [3], and Williams et al. [4]. Related applications and reviews may be found in Refs. [5], [6], [7], [8], [9], [10], [11].

Here, the theory governing the mechanical (and less so electrical) behaviour of such devices is examined in detail. Although the base excitation of a single degree of freedom (dof) spring–mass–damper model is perfectly well understood—it is described in most, if not all, typical undergraduate textbooks on vibration (see, for example Ref. [12])—many of the derived expressions can be written in a bewildering variety of forms, and it is shown how easy it is to draw erroneous conclusions, as some authors have done. Typically, these relate to the role of damping. The overall damping coefficient (of which the mechanical damping is a part) plays an ambiguous role, being associated not only with the harvesting and dissipation of energy, but also the transfer of energy from the vibrating base into the system. It is shown that net energy flow from the base through the damper is positive (negative) for ω>ωn (ω<ωn), but is zero when ω=ωn. It is also shown that the mechanical component of the overall damping cannot contribute more energy than it dissipates; this is consistent with experimental evidence by Williams et al. [4] indicating that the operation of such a device in a vacuum is beneficial. Potential design features of future devices are proposed. One must recognise that in any practical device, the amplitude of the seismic mass is limited to some maximum value and this, together with the magnitude and frequency of the excitation (the operating environment), define the maximum power that can be extracted from the environment. A highly damped system has the advantage of harvesting energy over a wider band of excitation frequencies on either side of the natural frequency, is potentially smaller, but will harvest marginally less power. The theory is developed first for an alternating force applied directly to the seismic mass, before considering the more involved case of base excitation; it transpires that while the dimensionless average powers for the two cases are different in form, their dependence on frequency and damping ratios is identical. It is noted that three different resonant frequencies can be defined for the two cases considered, but maximum power is extracted from the environment when the frequency of excitation is equal to the undamped natural frequency, irrespective of the damping ratio.

Having considered the effect of total damping—consisting of useful energy harvested, together with electrical and mechanical dissipation—the effect of these is considered individually. The concept of impedance matching for maximum power delivery to an electrical load is addressed. Again, various authors have made conflicting claims, some suggesting that the impedance of the electrical load should be matched to the source impedance of the generator (coil), others that power delivery is optimised when mechanical and electrical damping ratios are equal. It is shown that maximum power is delivered to the electrical load when its resistance is equal to the sum of the coil internal resistance and the electrical analogue of the mechanical damping coefficient (here such matching is termed EDAM). Practically, this is equivalent to the matching of electrical and mechanical damping, if the coil resistance is small. However, mechanical damping should be as small as possible, which defines a minimum electrical damping coefficient dependent on the size of the device.

A possible strategy to harvest energy from a variable amplitude environment is also discussed, suggesting that controllable electrical damping is advantageous, so the ideal sensor would be intelligent with regard to both the harvesting of the energy, and its subsequent use.

Last, it is noted that the analysis is directed towards present generation devices that may be loosely described as millimetrecentimetre sized; however, as size reduction leads towards what may be described truly as MEMS [13], where characteristic lengths are typically less than 1 mm but greater than 1 μm (and are fabricated using integrated-circuit batch-processing technologies), miniaturisation has the effect of increasing the ratio of surface area (length2) to mass (length3) so surface effects, typically viscous forces, become more important. This suggests the reduction of mechanical damping by operating such a device in partial vacuum; in turn, one can envisage the breakdown of continuum viscous assumptions, and the necessity of stochastic descriptions (the Langevin equation) for the dynamics of the seismic mass. Such issues are not addressed here.

Section snippets

Direct mass excitation

The 1dof spring–mass–damper shown in Fig. 1 is subject to a sinusoidal force applied directly to the mass; the governing equation of motion is [12]mx¨+cx˙+kx=Fsinωt;dot denotes differentiation with respect to time. Multiply by velocity x˙, and rearrange asx˙Fsinωt=c(x˙)2+mx˙x¨+kxx˙orx˙Fsinωt=c(x˙)2+ddt(mx˙22+kx22).This represents conservation of power: in words, Eq. (2) states that the instantaneous power into the system is equal to the instantaneous power dissipated and/or absorbed by the

Electrical power generation

The arguments presented so far have made no real distinction between desirable damping—electrical energy harvesting from the device—and the inevitable mechanical and electrical losses. Also, they have focused on power flow from the environment into the device, rather than the delivery of useful power to an electrical load. These issues are now discussed in more detail for the case of base excitation. In common with most previous investigations, inductance is ignored. The simplest form of

Relationship with other published work

Despite claiming that damping ratio should be small in order to harvest the maximum amount of power, Williams and Yates [1] presented theoretical predictions of generated power for a range of frequencies of excitation, amplitude of vibration source, and seismic mass displacement (relative to the base) for a damping ratio ξ=0.3. In fact these represent power extracted from the environment, rather than power delivered to an electrical load, for the reasons stated above; the numerical values

Behaviour of an intelligent device

Now consider the possible behaviour of an intelligent device in response to variable amplitude base excitation: we suppose that the device has been designed for maximum power transfer into the electrical domain (ξe=ξm, or Rload+Rint=K2/cm) for some target excitation amplitude Y0, with a maximum seismic mass excursion, from Eq. (42), ofZmax=Y0/(4ξe);this defines the minimum electrical damping ratio. The (design) power flow into the electrical domain may be written asPdes=ce2(ce+cm)mωn3Y0Zmax=mωn3

Conclusions

The extraction of energy from a vibrating environment has been analysed in some detail. For both direct mass (force) and base excitation, the maximum power flow into the device depends on the vigour of the environment (frequency and amplitude of force, or base) and the size of the device. The dependence on frequency and damping ratios is the same for both cases, with maximum flow when the frequency of excitation is equal to the undamped natural frequency. A highly damped system would extract

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